Confined Plasticity in Micropillars

CHAPTER 9

Confined Plasticity in Micropillars Chapter Outline 9.1 Insights into Coated Micropillar Plasticity 9.1.1 9.1.2 9.1.3 9.1.4

240

Stress-Strain Curves in Coated and Uncoated Pillars 240 Dislocation Source Mechanism in Coated Micropillars 241 Back Stress in Coated Micropillars 244 Evolution of Mobile and Trapped Dislocation 245

9.2 Implications for Crystal Plasticity Model 247 9.3 Theoretical Models for Coated Micropillars 252 9.3.1 Dislocation Density Evolution Model 253 9.3.2 Prediction of Stress-Strain Curve 255

9.4 Brief Discussion on Coating Failure Mechanism 257 9.4.1 High Hoop Stress of Coated Layer 258 9.4.2 Transmission Effect of Dislocations Across Coating

9.5 Summary

258

261

Interfaces are commonly introduced to improve the strength of the material, because interfaces can partly block dislocation motion. Examples are found in bimaterials, polycrystals with grain boundaries (Gao et al., 2011b; Li et al., 2009b; Quek et al., 2016) or twin boundaries (Fan et al., 2015), and materials with a coating interface. However, in most cases, enhanced strength is accompanied by a loss of ductility. Therefore, it is necessary to understand confined plasticity to design strong and reliable materials. In this chapter, we take the coated micropillar as an example to study confined plastic behavior, because crystal devices in microelectromechanical systems (MEMS) often have a protective hard coating. This gives them higher strength but also greatly improves their erosion or wear resistance, prevents stiction or electrical shorting (Hoivik et al., 2003), and improve thermal stability (Zhuang et al., 2006). However, the deposition of coating also leads to ultrahigh local flow stress during plastic deformation (Greer, 2007; Gu and Ngan, 2012; Jennings et al., 2012; Lee et al., 2013; Ng and Ngan, 2009a) and causes mechanical reliability issues for MEMS. This chapter aims to reveal the underlying dislocation mechanism and predict flow stress in confined plasticity at the microscale. It is organized as follows. Section 9.1 presents the mechanical behavior and underlying dislocation mechanisms in coated pillars and compares them with the uncoated counterpart, mainly based on dynamic causal modeling Dislocation Mechanism-Based Crystal Plasticity. https://doi.org/10.1016/B978-0-12-814591-3.00009-1 Copyright © 2019 Tsinghua University Press. Published by Elsevier Inc.

239

240 Chapter 9 (DCM) simulations. Then, Section 9.2 preliminarily connects the insights obtained by simulation to the continuum crystal plasticity theory for confined plasticity at a microscale. In Section 9.3, a simple stochastic theory model is described that can predict the upper and lower bounds of flow stress in the coated micropillar. Finally, the coating failure mechanism is preliminarily discussed in Section 9.4, including high hoop stress and the transmission effect of dislocation from the interface.

9.1 Insights into Coated Micropillar Plasticity Compression test experiments for coated micropillars provided a good opportunity to investigate the confined plasticity problem. During compression, numerous dislocations were trapped at the pillar-coating interface. The following discussion compares coated and uncoated micropillars based on a series of DCM simulations and experimental results, to give a clear picture about the coating effect on microscopic deformation mechanisms and mechanical responses. Some theoretical models are discussed based on insights into coated micropillar plasticity.

9.1.1 Stress-Strain Curves in Coated and Uncoated Pillars By carrying out compression tests for Au pillars with diameters of 500e900 nm, Greer (2007) first reported that the coated pillars displayed much higher flow stress and a significant amount of linear strain hardening, which differed substantially from those of pillars with free surfaces. Then, Ng and Ngan (2009a) pointed out that the overall mechanical response was insensitive to the volume fraction of the coating Vcoating, when Vcoating varied from 0.07 to 0.32. These results suggested that the load-sharing effect was unimportant in the coated pillars under consideration. Fig. 9.1A gives typical DCM simulation results of an Ni single crystal micropillar with and without an Al2O3 coating (Cui et al., 2015a). The pillar diameters range from 200 to 800 nm. The thickness of the Al2O3 coating layer is 5 nm as measured in the experiments (Lee et al., 2013). For uncoated pillars, the stress-strain curves exhibit no evident strain hardening, as shown in Fig. 9.1. For coated pillars, linear strain hardening and higher stress are observed (Greer, 2007; Jennings et al., 2012). The stress-strain curves for both uncoated and coated pillars are composed of strain bursts under constant stress, separated by elastic segments with a slope similar to the elastic modulus (Fig. 9.1A). The difference is that the strain burst is larger for uncoated than for coated pillars. Similar inhibited strain burst behaviors are also observed in experiments. Ng and Ngan (2009a) found that stress-strain behavior could be smoothened by coating, and strain bursts were effectively suppressed for micropillars with a diameter ranging from 1.2 to 6.0 mm and a Vcoating larger than about 0.26. Experimental results by

Confined Plasticity in Micropillars (A)

(B)

2000

Engineering stress (MPa)

241

d=200nm d=400nm 1600 d=600nm d=800nm

Uncoated Uncoated Uncoated Uncoated

Coated Coated Coated Coated

1200

800

400

0 0.0

0.2

0.4 0.6 0.8 Engineering strain(%)

1.0

Figure 9.1 Comparison of typical stress-strain curves between uncoated and coated micropillars (A) obtained by dynamic causal modeling simulations, and (B) obtained by experiments (Jennings et al., 2012). The dashed line in (A) reflects the purely elastic response. Reprinted from Cui, Y.N., Liu, Z.L., Zhuang, Z., 2015a. Theoretical and numerical investigations on confined plasticity in micropillars. Journal of the Mechanics and Physics of Solids 76, 127e143, Copyright 2015, with permission from Elsevier.

Jenning et al. (2012) further showed that the coating could not fully suppress the strain burst in small pillars with a diameter of 200 nm and Vcoating of about 0.17, and plastic strain recovery occurred during unloading. El-Awady et al. (2011) simulated the dislocation penetration process based on Koehler barrier strength criteria and discussed the influence of barrier strength on strain burst. The initiation of large strain bursts was delayed to higher strain and stress regimes with an increase in the barrier strength. In addition, as the barrier strength increased, the magnitude of the strain bursts decreased.

9.1.2 Dislocation Source Mechanism in Coated Micropillars As described in Chapter 8, for uncoated micropillars, a single-arm dislocation source (SAS) mechanism can well explain the mechanical response: the size effect is related to the characteristic length of the SAS, the intermittent strain burst is directly caused by the operation and shutdown of the SAS, and the lack of strain hardening is caused by the continuous operation of the stable SAS and weak dislocation interactions (Cui et al., 2014; El-Awady et al., 2009a,b; Parthasarathy et al., 2007). However, whether the SAS mechanism still works for coated micropillars needs to be restudied because a lot of dislocations are observed to pile up at the pillar-coating interface (Zhou et al., 2010b). By observing the evolution of dislocation microstructures, the researchers found that the plastic deformation of coated micropillars is mainly accommodated by the spiral motion of the SAS for the chosen dislocation density range (r0 > 47
1012m2) and deformation

242 Chapter 9

Figure 9.2 With the engineering strain at about 1%: (A) an overlay of images showing the operation of the single-arm dislocation source in the uncoated micropillars with a diameter of 400 nm; (B) The dislocation configuration showing the inner dislocation source character in coated micropillars with a diameter of 400 nm. Reprinted from Cui, Y.N., Liu, Z.L., Zhuang, Z., 2015a. Theoretical and numerical investigations on confined plasticity in micropillars. Journal of the Mechanics and Physics of Solids 76, 127e143, Copyright 2015, with permission from Elsevier.

stage (ε < 1.2%). One example is presented in Fig. 9.2. Dislocation configurations at 1% engineering strain are given for uncoated and coated pillars with a diameter of 400 nm, which have the same initial dislocation configuration. In the uncoated pillars, as shown in Fig. 9.2A, the stable SAS can continuously sweep the slip plane, generating a large amount of plastic strain. However, in the coated pillars, operation of the SAS leads to significant deposition of trapped dislocations, as shown in Fig. 9.2B. The SAS can operate only intermittently owing to back stress induced by the trapped dislocations. The initial dislocation density, r0, has an important role in influencing the destiny of the internal dislocation sources (Benzerga, 2009; El-Awady et al., 2013; Rao et al., 2008; Zhou et al., 2011). For coated micropillar with a low r0, the exhaustion of SAS may occur because high stress in the coated pillars may easily destroy the internal SAS. To analyze the coating effect on the activation of SAS quantitatively, the operation processes of individual strong SAS with indestructible pinning points in uncoated and coated micropillars are further presented in Fig. 9.3 (Cui et al., 2015a), which excludes the influence of collective dislocation interactions. Similar to the results with a complex dislocation configuration, the stable SAS in an uncoated sample can operate continuously once the applied stress reaches its operation stress. Nevertheless, in a coated sample, stress needs to increase intermittently to remobilize the SAS, leading to a high strain hardening rate (SHR) close to the elastic modulus, as shown in Fig. 9.3A. In addition, these results

Confined Plasticity in Micropillars (A)

(B)

243

(C)

Resolved shear stress (MPa)

600

450

d=400nm uncoated pillar coated pillar L

300 205 150 118 0 0.0

Z 0.5

1.0

1.5

Shear strain(%)

2.0 Y X

Figure 9.3 (A) Stress-strain curve for micropillar containing individual single-arm dislocation source under compression; (B and C) Dislocation configurations when shear strain is 2.45% in uncoated and coated micropillars, respectively. The dashed arrow in (C) indicates the characteristic length in the continuum back-stress model. Reprinted from Cui, Y.N., Liu, Z.L., Zhuang, Z., 2015a. Theoretical and numerical investigations on confined plasticity in micropillars. Journal of the Mechanics and Physics of Solids 76, 127e143, Copyright 2015, with permission from Elsevier.

show that for a free pillar, the operation of one stable source is enough to keep stable plastic flow, but for a coated pillar, observable plastic flow requires multiple sources to be activated simultaneously. As described in Section 8.4.3, critical resolved shear stress s to activate the SAS in uncoated samples can be estimated as (Lee and Nix, 2012; Parthasarathy et al., 2007) pffiffiffi ks m s ¼ s0 þ amb r þ l=b

(9.1.1)

where s0 is the lattice friction stress, b is the magnitude of the Burgers vector, r is the dislocation density, and l is the effective SAS length. Dimensionless constants a and ks are set to 0.5 and 1.0, respectively (Lee and Nix, 2012; Parthasarathy et al., 2007). The simulation results in Fig. 9.3A show that the critical resolved shear stress to activate an SAS in an uncoated micropillar is 118 MPa. For the considered case, the second dislocation interaction term on the right side of Eq. (9.1.1) can be ignored because there is no evident forest hardening. The deduced coefficient ks is very close to 1.0 according to Eq. (9.1.1). In uncoated samples, the operation stress of the SAS depends on the effective source length but also the angle between the initial dislocation segment and the Burgers vector (Rao et al., 2007). Thus, only in a statistical sense is the value of ks 1.0 for an uncoated sample.

244 Chapter 9 To extend this model to predict the operation stress of the SAS in the coated case, coefficient ks should be recalibrated to take into account the pinning effect of coating. Meanwhile, the back-stress term sb should be introduced: pffiffiffiffiffiffiffiffiffiffiffiffiffi ks m þ sb s ¼ s0 þ amb rmobile þ l=b

(9.1.2)

where rmobile represents mobile dislocation density. Why rmobile is used instead of r will be discussed later. Simulations indicate that the SAS in coated micropillars is similar to the Frank-Read (FR) source, and its operation stress is insensitive to the geometrical orientation. Thus, coefficient ks for the coated case can be estimated by the simulation results. According to the critical resolved shear stress that initially activates the SAS, sy ¼ 205 MPa (sb ¼ 0), as shown in Fig. 9.3A, it can be deduced that ks z 2.0. The estimation of back stress sb is given and expressed as follows.

9.1.3 Back Stress in Coated Micropillars For the individual strong SAS considered in Fig. 9.3, rmobile and l are substantially unchanged during deformation. Thus, the sum of the first three terms on the right side of Eq. (9.1.2) is equal to the initial operation stress sy ¼ 205 MPa. Accordingly, the back-stress term can be obtained by sb ¼ s$Msy, where M is the Schmidt factor. The relations between back stress and instantaneous trapped dislocation density rtrapped are plotted in Fig. 9.4A. The back stress increases stepwise as the pileup of dislocations. (A)

(B) Dimensionless constant

d=400nm

400

Simulation results Linear fit line

τb (MPa)

300

200

τb /ρtrapped ≈ α '. μ bd

2.4

Calculated ks

ks=2.0

2.1

Calculated α'

α'=0.7

1.8 1.5 1.2 0.9

100

0.6 0

0

20

40

ρtrapped (1012m-2)

60

80

200

400 600 Diameter (nm)

800

Figure 9.4 (A) Back stress versus trapped dislocation density for micropillars with a diameter of 400 nm; (B) Calculated values of ks and a0 for micropillars with different diameters. Reprinted from Cui, Y.N., Liu, Z.L., Zhuang, Z., 2015a. Theoretical and numerical investigations on confined plasticity in micropillars. Journal of the Mechanics and Physics of Solids 76, 127e143, Copyright 2015, with permission from Elsevier.

Confined Plasticity in Micropillars

245

If ignoring these discrete stress steps, a linear relation can be found between the trapped dislocation density and back stress. In a micropillar with diameter d, the mean diameter of dislocation loops located at the pillar-coating interface can be approximately taken as d. According to the classical dislocation pileup theory, the local resolved back stress acting on a given dislocation source resulting from the n already-emitted dislocation loops can be estimated by sb z npmb/d (Allain and Bouaziz, 2010; Hirth and Lothe, 1982). This implies that the back stress linearly depends on the trapped dislocation density, because it also linearly depends on the number of dislocation loops, n. Besides, the linear relation between flow stress and geometrically necessary dislocation (GND) was also suggested by Guruprasad and Benzerga (2008a). Accordingly, it is reasonable to assume that the following linear relation exists in a dimensionless form: sb =mza0 $rtrapped $b$d

(9.1.3)

Here, coefficient a0 is approximately 0.7, according to the least-square fitting of simulation results in Fig. 9.4A. To verify that ks and a0 are constants independent of the sample size, similar simulations were carried out in coated micropillars with other sizes. They also initially contained one individual SAS with the same slip plane, Burgers vector, effective length, and initial orientation. In coated micropillars with a diameter of 200, 600, and 800 nm, critical resolved shear stress to operate the SAS was 399, 142, and 110 MPa, respectively. The corresponding ks was calculated as shown in Fig. 9.4B. By least-square fitting of sb and rtrapped data, the values of a0 were also obtained. Fig. 9.4B shows that both ks and a0 are almost independent of the sample size. Then, by combining Eqs. (9.1.2) and (9.1.3), the activation stress of the SAS in coated samples can be estimated as pffiffiffiffiffiffiffiffiffiffiffiffiffi ks m s ¼ s0 þ amb rmobile þ þ a0 rtrapped bdm l=b

(9.1.4)

9.1.4 Evolution of Mobile and Trapped Dislocation Obviously, a prerequisite for applying Eq. (9.1.4) is to obtain the evolution law of mobile and trapped dislocation density. In the following discussion, the simulation results for micropillars with different complex dislocation configurations are further analyzed to study the evolution of rmobile and rtrapped. The distinction method between rmobile and rtrapped in DCM simulations is described in detail in Cui et al. (2015a). The evolution of dislocation density in a pillar with a diameter of 400 nm is illustrated in Fig. 9.5. For the coated pillar,

246 Chapter 9

Dislocation density(1012m-2)

400 Coated case

ρ

Uncoated case

ρ

300

total

ρ

trapped

ρmobile

total

200

100

0 0.0

0.2

0.4 0.6 0.8 Plastic shear strain(%)

1.0

Figure 9.5 Evolution of total dislocation density rtotal, mobile dislocation density rmobile, and trapped dislocation density rtrapped in an uncoated and coated micropillar with a diameter of 400 nm. Reprinted from Cui, Y.N., Liu, Z.L., Zhuang, Z., 2015a. Theoretical and numerical investigations on confined plasticity in micropillars. Journal of the Mechanics and Physics of Solids 76, 127e143, Copyright 2015, with permission from Elsevier.

though, the total dislocation density increases during deformation, and the mobile dislocation density first decreases and then tends to remain stable, similar to the uncoated case. Generally, mobile dislocations exist in two forms. One is surface dislocation with both ends terminating at the surface/interface. The other is dislocation sources with one or two anchor points inside the pillar resulting from the formation of dislocation junctions or jogs. The dislocation microstructures observed in the simulations reveal that most surface dislocations quickly glide to the interface region and pile up during the initial microplastic stage. The main form of the mobile dislocations is the SAS. Thus, to a certain extent, the initial gradual decrease in mobile dislocation density reflects the gradual destruction of the SAS. On the other hand, the trapped dislocation density has a linear dependence on the plastic strain, as indicated in Fig. 9.5. Because the plastic deformation is mainly induced by the operation of the SAS, it is natural to think of deriving the evolution of the trapped dislocation density by an SAS model. When the SAS rotates one circle, slipped area Aslip is pd(d/cos b)/4 (Fig. 8.9). Assuming the slip plane is swept by the SAS n times, the produced plastic shear strain, gp, can be calculated as  nbAslip nbpd 2 ð4cos bÞ nb (9.1.5) ¼ ¼ gp ¼ 2 pd h=4 hcos b V where V is the sample volume and h is the height of the sample. Meanwhile, the accumulated dislocation length corresponding to each deposited dislocation loop by the

Confined Plasticity in Micropillars

247

dρtrapped /dγp (1015m-2)

100 Crystal plasticity model prediction SAS model prediction Simulation results

80

60

40

20

0

200

400

600

800

Diameter (nm)

Figure 9.6 Accumulation rate of trapped dislocation for different diameters. SAS, single-arm dislocation source. Reprinted from Cui, Y.N., Liu, Z.L., Zhuang, Z., 2015a. Theoretical and numerical investigations on confined plasticity in micropillars. Journal of the Mechanics and Physics of Solids 76, 127e143, Copyright 2015, with permission from Elsevier.

SAS operation is p(d þ d/cos b)/2. Therefore, trapped dislocation density rtrapped after n cycles of rotation of the SAS is rtrapped ¼

npðd þ d=cos bÞ=2 pd 2 h=4

(9.1.6)

By substituting Eq. (9.1.5) into Eq. (9.1.6), we have rtrapped ¼

2ð1 þ cos bÞ gp bd

(9.1.7)

Eq. (9.1.7) shows the linear relationship between rtrapped and gp. On the other hand, the linear coefficient between rtrapped and gp can be calculated by fitting rtrapped and gp data in Fig. 9.5. The initial small plastic strain stage (gp ¼ 0e0.02%) is not taken into account in linear fitting because plastic deformation at this stage is accommodated by inner dislocation glide and dislocation configuration rebuilding without significant interface dislocation deposition. The simulation results for different pillar diameters are fitted and plotted together with SAS model predictions in Fig. 9.6. The coefficients are well-predicted by Eq. (9.1.7). The accumulation rate of trapped dislocation drtrapped/dgp decreases as the pillar diameter increases.

9.2 Implications for Crystal Plasticity Model Within the realm of continuum mechanics, the gradient crystal plasticity theory is often applied to investigate confined plasticity at the microscale. In the strain gradient theory,

248 Chapter 9 the expression of GND and its influence on strain hardening behavior have been extensively studied (Bayley et al., 2006). One typical work is the higher-order crystal plasticity theory developed by Gurtin (2002), in which back stress associated with trapped dislocations is incorporated. The back stress is derived from a defect energy term, which is the quadratic function of GND. This model can capture the size effect successfully in several constraint plastic flow problems (Bittencourt et al., 2003; Gurtin et al., 2007; Nicola et al., 2005a). However, the physical meaning of defect energy and the length parameters in the theory are not clear. For the coated pillar during plastic deformation, the coating introduces a deformation gradient because the coating and micropillar have different mechanical properties. Therefore, the trapped dislocation can be considered a GND, which is automatically maintained to ensure deformation compatibility (Ohno and Okumura, 2007). In addition, it is always expected that the GND are concentrated near the interface (Cleveringa et al., 1997; Evers et al., 2004a). Accordingly, if the evolution of GND and back stress can be obtained by a lower-scale discrete dislocation simulation, the results can be used directly to develop the strain gradient plasticity theory. DCM simulation results show that the back stress and trapped dislocation density have a linear relation (Fig. 9.4), and so do the trapped dislocation density and plastic shear strain (Eq. 9.1.7). In this section, these results are further verified by comparing them with solutions obtained from higher-order crystal plasticity theory. One of the most difficult challenges in higher-order crystal plasticity theory is how to formulate flow stress through the dislocation density at a microscale. DCM simulation results are preliminarily explored in the hope of shedding some light on this problem. In the following discussion, the compression of the coated micropillar is analyzed by using the higher-order crystal plasticity model developed by Gurtin (2002). Here, to facilitate comparison with the simulation results given in Section 9.1, it is supposed that only one slip system denoted by “k” is activated. A local coordinate system, OXYZ, is built in the elliptic slip plane, as indicated in Fig. 9.7A, where the origin of the coordinate is the center of the ellipse. The base vectors of axes X, Y, and Z are e1, e2, and e3, respectively. Here, e1 and e2 are along the minor and major axis of the ellipse, respectively, so that e3 is the out-of-plane direction, which is equal to the unit vector indicating the slip plane normal m(k). Assume that in the kth slip system the angle between e1 and slip direction is f. Then, the slip direction s(k) can be expressed as sðkÞ ¼ cos fe1 þ sin fe2

(9.2.1)

The corresponding tangent line direction of edge dislocation I(k) can be expressed as IðkÞ ¼ mðkÞ
sðkÞ

(9.2.2)

Confined Plasticity in Micropillars (A)

249

(B)

A=πad/4 O

Z

Z O a /2

X

β Y

Figure 9.7 Schematic description of dislocations gliding in the slip plane in a coated micropillar (A) in full three-dimensional view, where the blue elliptical ring is the coating layer; and (B) in cross-sectional view. Reprinted from Cui, Y.N., Liu, Z.L., Zhuang, Z., 2015a. Theoretical and numerical investigations on confined plasticity in micropillars. Journal of the Mechanics and Physics of Solids 76, 127e143, Copyright 2015, with permission from Elsevier.

Confining attention to the small displacement gradient, the total strain rate ε_ can be decomposed as an elastic part ε_ e and a plastic part ε_ p : ε_ ¼ ε_ e þ ε_ p ; ε_ p ¼

NS X

ðkÞ ðkÞ g_ ðkÞ ¼ p P ; P

k¼1

 1  ðkÞ s 5mðkÞ þ mðkÞ 5sðkÞ 2

(9.2.3)

ðkÞ

where g_ p is the slip rate on the kth slip system, NS denotes the number of activated slip system (NS ¼ 1 here), and P(k) is the Schmid tensor. Stress rate tensor s_ is related to the elastic strain by s_ ¼ Ce : ε_ e

(9.2.4)

The microforce balance equation is expressed as s_ ðkÞ ¼ p_ ðkÞ  V$x_

ðkÞ

(9.2.5)

where s(k) is the resolved shear stress in the kth slip system. p(k) is the slip resistance, which ðkÞ corresponds to the first three terms on the right side of Eq. (9.1.2). V$x_ corresponds to

the back-stress term in Eq. (9.1.2). Higher-order stress x(k) can be expressed by the partial derivative of defect energy j(k) with respect to the slip gradient (Gurtin, 2002):   vjðkÞ ðkÞ 1 2 X  ðkÞ ðkÞ 2  ðkÞ ðkÞ 2 ðkÞ x ¼ Vgp $s (9.2.6) ; j ¼ S0 L þ Vgp $I ðkÞ 2 vVgp k

250 Chapter 9 where S0 is a material constant with the dimension of stress. L is a characteristic length, which is determined later. Because we restrict attention to the uniaxial compression test for a micropillar, only the axial components of stress and strain tensor, such as s, ε, and εp, are discussed next. According to Eqs. (9.2.3) and (9.2.4), axial stress rate s_ can be expressed as ! NS X  ðkÞ ðkÞ s_ ¼ E ε_  ε_ p ¼ E ε_  (9.2.7) g_ p M k¼1

By taking  only the kth  slip system into account, the axial stress rate can be expressed as ðkÞ _ Combined with Eqs. s_ ¼ E ε_  g_ p M ðkÞ . The resolved shear stress rate is s_ ðkÞ ¼ M ðkÞ s. (9.2.5) and (9.2.6), the general solution for the plastic shear strain rate can be obtained: . pffiffiffi pffiffiffi pffiffiffi pffiffiffi ðkÞ QY  QY QX  QX _ g_ ðkÞ ¼ ε M þ C e þ C e þ C e þ C e ; 1 2 3 4 p  2  (9.2.8) Q ¼ E M ðkÞ S0 L2 where C1, C2, C3, and C4 are constants to fulfill the boundary conditions. Here, the microclamped conditions at the coated interface are given by gðkÞ p ðX; YÞ ¼ 0; if

X2 ðd=2Þ

þ 2

Y2 ða=2Þ2

¼1

(9.2.9)

To obtain the analytical solution of Eq. (9.2.8), the first term is kept unchanged and Taylor series are used around original point O for the other terms, which is given by .   pffiffiffiffi pffiffiffiffi pffiffiffiffi 2 .  pffiffiffiffi 2 .  ðkÞ _ g_ ðkÞ ¼ ε M þ C 1 þ Q Y þ Q Y 2 þ C 1  Q Y þ QY 2 þ 1 2 p  pffiffiffiffi pffiffiffiffi  pffiffiffiffi 2  pffiffiffiffi 2 .  QX 2 QX 2 þ C4 1  Q X þ C3 1 þ Q X þ (9.2.10) According to the boundary conditions, the solution is obtained by  2 2 2 2 2 2 g_ ðkÞ p ¼ P 4d Y þ 4a X  a d ; P ¼

_εEM ðkÞ  2 8S0 L2 ða2 þ d2 Þ þ E M ðkÞ a2 d2

(9.2.11)

Because the mixed dislocations can be decomposed into edge and screw components, the corresponding GND density is calculated (Ohno and Okumura, 2007): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    ffi ðkÞ

r_ GND ¼

1 ¼ b

ðkÞ

r_ GND;edge

2

ðkÞ

þ r_ GND;screw

2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2ffi 8jPj pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðkÞ ðkÞ ðkÞ ðkÞ a4 X 2 þ d 4 Y 2 þ Vg_ p $I ¼ Vg_ p $s b

(9.2.12)

Confined Plasticity in Micropillars

251

Here, the dislocation cross-slip and climb are not taken into account. Assuming dislocation glide only, the pillar is considered to be composed of many elliptic cylinders, each of which corresponds to a slip plane. Thus, the volume-averaged GND is thought to be equivalent to the average GND density through the slip plane: Z 2p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  E 1Z  D 4jPja2 d  ðkÞ  ðkÞ _ r ¼ cos2 q þ cos2 b sin2 qdq (9.2.13) ; C ¼ C r_ GND ¼  dA 5 5 A A GND 3pb 0 where C5 is a dimensionless constant, which depends only on the loading orientation. Correspondingly, the average plastic shear strain caused by dislocation slip is  E 1Z  D jPja2 d 2  ðkÞ  _gðkÞ _ (9.2.14) ¼ gp dA ¼ p A A 2 According to Eqs. (9.2.13) and (9.2.14), the relation between the average GND density and average plastic shear strain can be deduced as D E 8C5 D ðkÞ E ðkÞ g_ r_ GND ¼ (9.2.15) 3pbd p E D E D E.D E D ðkÞ ðkÞ ðkÞ ðkÞ g_ p ¼ Eq. (9.2.15) reflects the linear dependence of r_ GND on g_ p , and r_ GND 4:2=ðbdÞ for the [001] loading orientation. As discussed previously, the trapped dislocation can be thought as GND. According to the SAS model prediction in Section 9.1.4, Eq. (9.1.7) also gives a linear relation between the trapped dislocation density and volumeaveraged plastic shear strain. For the slip plane considered here, drtrapped/dgp is 3.2/(bd). The relations predicted by both Eq. (9.1.7) and Eq. (9.2.15) agree well with the simulation results shown in Fig. 9.6. Furthermore, the average back stress can be obtained as Z   1  ðkÞ  h_sb i ¼   V$x_ dA ¼ 8S0 L2 jPj a2 þ d2 A A Combining Eqs. (9.2.16) and (9.2.13), the following relation can be derived: 

6p 1 þ cos2 b S0 L2 D ðkÞ E r_ GND bd h_sb i=m ¼ C5 md 2

(9.2.16)

(9.2.17)

Comparing Eq. (9.2.17) with Eq. (9.1.3), the characteristic parameter S0L2 can be determined to be 0.14md2. According to work by Liu et al. (2011), S0 ¼ m/8(1n), which is an elastic constant. Then, L is estimated as 0.88d, as denoted in Fig. 9.3C. The characteristic length L just corresponds to the size of the typical dislocation configuration that is influenced by the extrinsic characteristic length; here, d is the pillar diameter. This provides a reference for determining the length parameter in a higher-order back stress model. The material length parameter reflects the region of influence of short-range dislocation interactions.

252 Chapter 9 In gradient-based plasticity formulations, identification of a certain dislocation as being statistically stored or geometrically necessary remains unclear (Evers et al., 2004a). Guruprasad and Benzerga (2008a,b) reported some inspiring 2-5-dimensional (2.5D)discrete dislocation dynamics (DDD) work to analyze the local GND density in a free micropillar using the net Burgers vector based on Nye’s tensor. However, in 3D-DDD, a strict distinction of GND is difficult. Consistency between the crystal plasticity theory and simulation results here further suggests a correspondence between GND and trapped dislocation and a linear relation between back stress and trapped dislocation density, which may shed some light on intuitively understanding the GND. Many studies also consider the contribution of GND to slip resistance by the Taylor interaction term (Fleck et al., 1994; Han et al., 2005a). However, Mayeur and McDowell (2013) found that adding the GND density to the Taylor relation would overestimate flow stress. Because the back-stress term already considers the contribution of GND, including the GND density in the Taylor relation will double-count its contribution. The simulation results presented here show that the trapped dislocations (GND) contribute most to the flow stress and the trapped dislocation density increases linearly with plastic strain in the coated pillars. Therefore, if the contribution of trapped dislocation is introduced by the Taylor hardening law, a square root dependence of flow stress on plastic strain is obtained. However, linear strain hardening is clearly observed in both simulations and experiments (Greer, 2007) in coated pillars at the microscale. This means that the trapped dislocations contribute to an increase in flow stress in terms of back stress hardening instead of Taylor hardening. That is why only the back-stress term in Eqs. (9.1.3) and (9.1.4) considers the trapped dislocation density, which is also consistent with previous work (Guruprasad and Benzerga, 2008a). The slip system resistance is mainly influenced by mobile dislocations and the internal dislocation source operation. In particular, at such a small scale, the source operation has a more crucial role compared with the Taylor interactions in relation to the mobile dislocation density.

9.3 Theoretical Models for Coated Micropillars Some theoretical studies were also performed to develop a continuum model describing confined plasticity in coated pillars. Lee et al. (2013) used a simple numerical model to illustrate the coating effect on the source operation. He introduced an additional stress term, Dscoating, to the SAS model to consider dislocation pinning and the pile-up effect. However, the value of Dscoating was directly calculated from the experimental sample strength and its evolution was not provided. Thus, it was difficult to use it to predict the mechanical response for other samples. On the other hand, some researchers try to correlate high flow stress with the total dislocation density in coated pillars based on the Taylor hardening theory (Jennings et al., 2012; Ng and Ngan, 2009a). For a large coated pillar, a good correlation was obtained (Gu and Ngan, 2012). However, for a small coated

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pillar (w<1 mm), the Taylor relation failed. Generally, the Taylor relation worked well when forest dislocation hardening prevailed. Nevertheless, studies (Gu and Ngan, 2012) indicated that the coating could not store the forest dislocations effectively in a small sample. Internal mobile dislocations were scarce and most dislocations were trapped at the interface. In the following discussion, based on the operation stress equation of SAS and the linear back-stress model obtained from DCM simulations, a theoretical model is presented that can conveniently predict the stress-strain curve of a coated sample.

9.3.1 Dislocation Density Evolution Model From Fig. 9.1, it is reasonable to assume that for the considered sample size, all of the plastic strain is composed of discrete strain bursts and the stage between each strain burst is a pure elastic response. Accordingly, as long as the stress value at which strain burst occurs and the magnitude of strain burst are known, the stress-strain curve can be predicted. According to the simulation results in Section 9.1, the plasticity of the coated pillar is still controlled by the operation of the SAS, so each detectable strain burst is assumed to be caused by the operation of the SAS, similar to that in the uncoated pillar (Cui et al., 2014; Rao et al., 2008). Therefore, the key issue is to calculate the operation stress of the SAS and their evolution. First, given initial dislocation density r0 and the pillar’s geometrical size, the number of SAS can be estimated by n ¼ integer[rpdh/4], according to Eq. (8.4.9). Based on the statistical model described in Section 8.4.2, the statistically averaged effective length of the SAS can be estimated but only one slip system can be considered. Here, the dislocation activities for all the available slip systems are considered by a stochastic method inspired by Ng and Ngan’s work (2008a,b). We assumed that each SAS is randomly assigned to one of 12 slip systems in a face-centered cubic (FCC) crystal. The pinning point for each SAS distributes randomly in the corresponding elliptic slip plane, with the same probability of locating at any position in the slip plane. The shortest distance of each pinning point from the elliptic perimeter is calculated as the effective length of SAS l(j). Here, superscript “(j)” means the variables corresponding to the jth SAS. In the micropillar under consideration, the number of SAS is limited. For the early deformation stage (ε < 0.2%), let us assume that there is no interaction between SAS and no coupling between slip systems. The SAS can be activated one by one independently of each other. Thus, the stress plateau for each strain burst corresponds to the operation stress of one SAS, which can be deduced from Eq. (9.1.4) as " #, . k m ffiffiffiffiffiffiffiffiffiffiffiffiffi p s . þ 2a0 ð1 þ cos bÞgðjÞ sðjÞ ¼ sðjÞ M ðjÞ ¼ s0 þ amb rmobile þ M ðjÞ (9.3.1) p m lðjÞ b

254 Chapter 9 where Schmid factor M(j) for the jth SAS can be calculated according to its slip system information and the loading orientation. ðjÞ

Obviously, to obtain the evolution of s(j), the evolutions of rmobile, l(j), and gp must first be calculated. For simplicity, the change of rmobile is ignored because its influence on the ðjÞ source operation stress is weak compared with the other two terms in Eq. (9.3.1). gp can be calculated according to Eq. (9.1.5) by taking n as the operation times of the jth SAS. The most difficult one is to estimate the evolution of l(j), which reflects the magnitude of strain burst and corresponds to the lifetime of the SAS (i.e., the number of times that SAS can sweep the slip plane before it ruptures). In an uncoated micropillar, weak dislocation interactions and the small amount of SAS make it possible to predict the lifetime of the SAS statistically according to the instantaneous dislocation density and sample size (Cui et al., 2014). Nevertheless, strong interactions between the SAS and trapped dislocations in the coated case make it difficult to predict the lifetime of the SAS. For simplicity, two extreme cases are considered. From the simulation results shown in Fig. 9.3A, after the SAS sweeps the glide plane one time, back stress leads to the complete shutdown of this source. Therefore, one extreme case assumes that the SAS ruptures once it operates and sweeps the slip plane one time. Namely, l(j) become 0 in the subsequent steps after one operation. This can be considered the shortest lifetime of the SAS and the maximum effect of SAS exhaustion hardening. Therefore, the predicted mechanical response represents the upper bound of stress. The other assumes that all SAS have an unlimited lifetime and l(j) is unchanged during the whole deformation stage. This assumption considers back-stress hardening but ignores SAS exhaustion hardening. Thus, the predicted results represent a lower bound of stress. The physical process behind the extreme case that ignores the failure of the SAS is similar to the DDD simulation in which a population of FR sources is initially put in the crystal using indestructible pinning points and a presumed source length distribution (El-Awady et al., 2011; Zhou et al., 2010b). More details about the calculation procedure of this theoretical model are schematically described in Fig. 9.8. Under uniaxial compression, the applied stress elastically rises until the ath SAS with the minimum operation stress smin ¼ min(s(1), s(2), ., s(N)) activates. The operation stress of the weakest ath SAS can be considered the initial yield stress, which is also the stress value for the first strain burst. Accordingly, it begins from s ¼ s(a), ε ¼ s/E and ends with s ¼ s(a), ε ¼ s/EþgpM(a), where E is the Young’s modulus and gp is the plastic shear strain produced by the activated SAS, calculated according to Eq. (9.1.5) by setting n as 1. Afterward, the operation stress for the activated SAS needs to be updated. Then, the applied stress elastically increases again until another SAS with the minimum operation stress is activated. Correspondingly, the second strain burst occurs. This activated SAS also sweeps the glide plane one time and produces gp. This procedure is repeated until the expected loading is reached, as schematically described in Fig. 9.8. To plot the stress-strain curve, the stress and strain values are

Confined Plasticity in Micropillars Assign initial conditions, σ0 =0, ε0=0,ρ0, d, h, E, loading orientation.

λ(a) =0

Calculate SAS number, randomly distribute these SAS.

λ(a) is unchanged

Calculate the operation stress σ ( j), σmin=min(σ (1),…,σ (N) )

σ≥ σmin

Upper bound

Lower bound

Y

255

End Y N

Finish loading

The ath SAS operates. Update ρmobile,γ . Record results when strain burst starts and ends.

Figure 9.8 Calculation flowchart of theoretical model. SAS, single-arm dislocation source. Reprinted from Cui, Y.N., Liu, Z.L., Zhuang, Z., 2015a. Theoretical and numerical investigations on confined plasticity in micropillars. Journal of the Mechanics and Physics of Solids 76, 127e143, Copyright 2015, with permission from Elsevier.

recorded at the start and end of the ith strain burst, respectively. The ith strain burst starts when s2i ¼ smin and ε2i ¼ ε 2i-1 þ(s2ies 2i1)/E, and ends when s2iþ1 ¼ smin and ε2iþ1 ¼ ε2i þgpM (a), respectively.

9.3.2 Prediction of Stress-Strain Curve Some stress-strain responses for these two extreme cases obtained by the theoretical model prediction are given in Fig. 9.9A for the coated micropillar under uniaxial compression.

Figure 9.9 (A) Predicted upper-bound and lower-bound stress-strain curves for coated micropillar under uniaxial compression; (B) Comparison between theoretical model predictions and simulation results. The inset is a higher-magnification image of the microplasticity region during the early deformation stage. Reprinted from Cui, Y.N., Liu, Z.L., Zhuang, Z., 2015a. Theoretical and numerical investigations on confined plasticity in micropillars. Journal of the Mechanics and Physics of Solids 76, 127e143, Copyright 2015, with permission from Elsevier.

256 Chapter 9 Here, the loading axis is taken as [001] and the initial dislocation density is 1014 m2. The results for two extreme cases share some similarities: (1) The initial yield stresses are almost the same for the same diameter, although some slight differences arise from the stochastic distribution of sources. (2) The smaller micropillar exhibits higher flow stress, which agrees with the simulation results shown in Fig. 9.1A. Interestingly, although the stress-strain curves are composed of collective discrete strain bursts, the obtained stress-strain curves are smooth for large micropillars. This is consistent with a previous investigation (Csikor et al., 2007). Moreover, for a micropillar with a diameter of 200 nm, the hardening moduli for both extreme cases are close to the Young’s modulus, in accord with simulation results shown in Fig. 9.1A and previous experimental data in Fig. 9.1B (Jennings et al., 2012). The theoretical predictions and simulation results were directly compared for the pillar with a diameter of 400 nm, as shown in Fig. 9.9B. Here, the initial dislocation density for simulation results was close to 1014 m2. The results of the theoretical model were the average over 100 separate realizations to give a sense to the statistic representation. The strain hardening stage of the simulation results are well-captured by the theoretical model, whereas stress from the theoretical model predictions at the early stage of loading is higher than that in the simulation results. This is because the theoretical model assumes that the initial deformation is purely elastic, whereas in the simulations, microplasticity usually occurs owing to dislocation motion and the breakup of weak junctions (Motz et al., 2009; Ni et al., 2017), as shown in the inset of Fig. 9.9B. Moreover, the existence of microplasticity and the stochastic strain burst cause difficulty in defining the initial yield stress (Ispa´novity et al., 2010) for the simulation results. Hence, the yield stress is not compared in the simulation and theoretical model predictions. In the following discussion, the predicted strain hardening behavior is compared with the simulation results. The SHR is calculated by SHR ¼ dsT =dεT ðs > sc Þ; sT ¼ sð1 þ εÞ; εT ¼ lnð1 þ εÞ

(9.3.2)

where sT and εT are the averaged true stress and true strain, respectively, which are obtained according to engineering stress and strain. The averaged SHR is derived by leastsquare fitting the sT-εT curve. For the theoretical model, cutoff stress sc is the critical stress that activates the weakest SAS. For the simulation results, sc is taken to be the stress when the first detectable strain burst occurs (the burst extent Dε > 0.02%). The analytical prediction and simulation results for SHR are plotted in Fig. 9.10, in which the initial dislocation densities r0 used in the simulations are indicated. The simulation results fall into the region bounded by the upper and lower bounds. The SHR values exhibit some kind of size dependence. The smaller the sample size is, the higher the SHR is. In addition, the simulation results indicate that the micropillar with a

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Figure 9.10 Comparison of strain hardening rate (SHR) between theoretical model predictions and simulation results. The initial dislocation density r0 for the simulated micropillar is numerically labeled. The r0 for analytical results is 100 mm2. Reprinted from Cui, Y.N., Liu, Z.L., Zhuang, Z., 2015a. Theoretical and numerical investigations on confined plasticity in micropillars. Journal of the Mechanics and Physics of Solids 76, 127e143, Copyright 2015, with permission from Elsevier. *Results marked are from previous studies Lee, S.W., Jennings, A.T., Greer, J.R., 2013. Emergence of enhanced strengths and Bauschinger effect in conformally passivated copper nanopillars as revealed by dislocation dynamics. Acta Materialia 61, 1872e1885; Zhou, C., Biner, S., LeSar, R., 2010b. Simulations of the effect of surface coatings on plasticity at small scales. Scripta Materialia 63, 1096e1099.

lower initial dislocation density exhibits a higher SHR because it is easier for the SAS to exhaust, which is consistent with previous studies (Benzerga, 2009; Gu and Ngan, 2012). On the other hand, the activation of cross-slip is thought to have an important role in promoting dislocation multiplication and decreasing the SHR (Zhou et al., 2010b). However, the effect of cross-slip is not considered in either theoretical analysis or numerical simulations here. To discuss this, the SHRs are calculated according to simulation results in the available references that consider cross-slip, e.g., uniaxial compression tests on a Cu micropillar coated with TiO2/Al2O3 with a diameter of 200 nm and an initial dislocation density of about 100 mm2 (Lee et al., 2013), and coated Ni with a diameter of 500/1000 nm and an initial dislocation density of about 10e20 mm2 (Zhou et al., 2010b), respectively. As evidenced by the data in Fig. 9.10, the incorporation of cross-slip does not dramatically decrease the SHR for such a small coated micropillar.

9.4 Brief Discussion on Coating Failure Mechanism According to experimental observations, a coated micropillar is usually damaged by failure of the coating followed by delamination at the interface (Jennings et al., 2012). Failure of the coating may occur in two different ways. One is as a result of local high stress.

258 Chapter 9 The other is caused by penetration of the dislocations. These two mechanisms are preliminarily discussed next (Cui, 2016).

9.4.1 High Hoop Stress of Coated Layer Previous postdeformation scanning electron microscopy observations show that the coating usually fails as a result of axial cracks in the coating (Jennings et al., 2012). Thus, high hoop stress may be the reason for coating failure. As shown in Fig. 9.11A, lots of dislocations are trapped along the circumference of the pillar. The corresponding hoop stress distribution is obtained by DCM simulation, as shown in Fig. 9.11B and C. Even when the trapped dislocation density is not high, the hoop stress value of the coating is significantly higher than that of the micropillar. Therefore, it is easy for the coating to crack as a result of high local hoop stress. Then, a large deformation is triggered where the crack occurs in the coating layer (Ng and Ngan, 2009a).

9.4.2 Transmission Effect of Dislocations Across Coating On the other hand, a ceramic coating is opaque to dislocations and acts as a dislocation sink. Because of its limited ability to absorb matrix dislocations, the transmission of dislocation can lead to brittle-type failure of the coating. Whether a dislocation segment (A)

(B)

(C)

Hoop stress(GPa) (Avg: 75%)

Hoop stress(GPa) SNEG, (fraction = -1.0) (Avg: 75%)

+4.00e-01 +3.16e-01 +2.32e-01 +1.48e-01 +6.42e-02 -1.98e-02 -1.04e-01 -1.88e-01 -2.72e-01 -3.56e-01 -4.40e-01 -5.24e-01 -6.08e-01

Z

Z

Z

X

+7.75e-01 +4.00e-01 +3.17e-01 +2.34e-01 +1.52e-01 +6.87e-02 -1.41e-02 -9.69e-02 -1.80e-01 -2.63e-01 -3.45e-01 -4.28e-01 -5.11e-01 -5.94e-01

Y

X

Y

X

Y

Figure 9.11 A micropillar with a diameter of 400 nm in which the engineering strain is 0.6%: (A) The corresponding dislocation configuration, in which the blue dislocation lines are trapped and the others are the mobile; (B) a longitudinal cross-sectional view of hoop stress in the micropillar; and (C) a longitudinal cross-sectional view of hoop stress in the coating (Cui, 2016). SNEG is used to describe the orientation of surfaces in ABAQUS software.

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will pile up or transmit across the interface mainly depends on the stress acting upon it. As discussed earlier, when the dislocation line approaches the interface, it will be strongly repelled by the image forces. The dislocation transmission can occur only when the other stress acting on the dislocation overcomes the high image force. In DCM methods, it is difficult to calculate the stress field of dislocations close to the interface with accuracy because very fine meshes are needed to capture the singularity. To estimate the critical image stress value at which the transmission happens, the analytical solution for the image force of the bimaterial is used. The available analytical solution has been found only for some specific cases such as an edge-screw dislocation near a surface layer or a dislocation loop in an anisotropic biomaterial (Chu et al., 2012; Head, 1953a,b; Weeks et al., 1968). Generally, the analytical image force diverges as the distance between the dislocation and the interface nears 0. However, this divergence is an artifact of treating the dislocation as an infinitely thin filament. When the distance to the interface is equal to the core radius, r ¼ 2b, the repulsion reaches the maximum possible value (Koehler, 1970). For the cases considered here, the resolved shear stress for image force sc is 0.7e0.8 GPa. Thus, sc is taken to be 0.8 GPa in our simulation. To study the effect of the sc value, sc is also calculated when the cutoff radius is taken to be the magnitude of the Burgers vector. Here, sc ¼ 1.4e1.7 GPa and sc is set as 1.5 GPa during the simulation. Then, dislocations can transmit if the following criterion is met: b$ðs þ sself þ sinter Þ$n > sc

(9.4.1)

where sself is the line tension, sinter is the interaction stress caused by the other dislocations, and n is the normal direction vector of the slip plane. Such a treatment is consistent with the dislocation penetration criteria (El-Awady et al., 2011). sc corresponds to the Koehler barrier strength (Koehler, 1970), which depends on the thickness of the coating and the material properties of the micropillar and coating layer. To investigate the dislocation transmission process and exclude the influence of collective dislocation interactions, the simulated samples contain only one SAS with indestructible pinning points. The parameters for the SAS are the same as those described in Section 9.1.3. Fig. 9.12A indicates that the strain hardening behavior can practically vanish when dislocation penetration is allowed, even if there is only one activated SAS. The final stable flow stress scales proportionally to the value of sc. The stable resolved shear stress is about 509 and 670 MPa when sc is 0.8 and 1.5 GPa, respectively. This observation is consistent with previous simulation work (El-Awady et al., 2011). The dislocation configurations when the engineering strain is 1% are shown in Fig. 9.12B and C, in which the local penetration sites are denoted by arrows. The hoop stress distributions are shown in Fig. 9.13, in which the dislocations begin to transmit from the interface. In the case of sc ¼ 800 MPa, the maximum hoop stress in the coating smax hoop is 588 MPa, as indicated by the gray region. In the case of sc ¼ 1500 MPa,

260 Chapter 9

Resolved shear stress (MPa)

(A)

(B)

800

(C)

670 600 509 400 d=400nm,coated pillar 200

τc =800MPa τc =1500MPa

0

0

1

2

3

4

Z

Shear strain(%)

Y X

Figure 9.12 (A) Stress-strain curve for micropillar containing individual single-arm dislocation source under compression; (B and C) Dislocation configuration when shear strain is 2.45% in coated micropillar with sc of 800 and 1500 MPa, respectively (Cui, 2016).

Figure 9.13 For a micropillar with a diameter of 400 nm, hoop stress of the coating is shown when dislocations begin to transmit from the interface (A) in the case of sc ¼ 0.8 GPa; and (B) in the case of sc ¼ 1.5 GPa (Cui, 2016). SNEG is used to describe the orientation of surfaces in ABAQUS software.

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smax hoop is 1090 MPa. Because the tensile strength of Al2O3 is only about 267 MPa (Munro, 1997), the coating will crack as a result of high hoop stress before dislocation transmission.

9.5 Summary In this chapter, the stress-strain behavior and underlying dislocation mechanisms of coated micropillars are presented to understand confined plasticity at the microscale. The evolution of mobile and trapped dislocations is separately discussed. The exhaustion of mobile dislocation reflects the destruction of the SAS; it contributes to flow stress through the Taylor hardening law. The trapped dislocation density exhibits a linear relation with plastic strain and induces an increase in flow stress in terms of back-stress hardening. Back stress contributes most to flow stress and exhibits a linear dependence on the trapped dislocation density. This relation correlates well with the derivation obtained by the higher-order crystal plasticity theory and is used directly to determine the material parameters in the continuum back-stress model. The theoretical model of predicting the upper and lower bounds of the stress-strain curve for coated compression micropillars is given. Finally, the basics of two kinds of coating failure mechanisms are discussed.